Is there any specific criteria for a matrix $A(t)$ to have either time-dependent or independent eigenvectors?

Question

I am investigating properties of a matrix

$$A(t_1,t_2) \equiv U_1(t_1) \otimes U_2(t_2) - U_2(t_2) \otimes U_1(t_1)$$

where $U_1$ and $U_2$ are time-dependent unitary matrices. I'm finding that for some choices of $U_1$ and $U_2$, when $t_1 \neq t_2$, the eigenvectors of $A$ are time-independent. Are there any testable properties of $A, U_1, U_2$ which could be used to predict this?